Theorem signslema 29456

Description: Computational part of signwlemn . (Contributed by Thierry Arnoux, 29-Sep-2018.)

Hypotheses

Ref	Expression
signslema.1	|- ( ph -> E e. NN0 )
signslema.2	|- ( ph -> F e. NN0 )
signslema.3	|- ( ph -> G e. NN0 )
signslema.4	|- ( ph -> H e. NN0 )
signslema.5	|- ( ph -> ( E < G /\ -. 2 || ( G - E ) ) )
signslema.6	|- ( ph -> ( ( H - G ) - ( F - E ) ) e. { 0 , 2 } )

Assertion

Ref	Expression
signslema	|- ( ph -> ( F < H /\ -. 2 || ( H - F ) ) )

Proof of Theorem signslema

Step	Hyp	Ref	Expression
1		signslema.5	. . . . . 6 |- ( ph -> ( E < G /\ -. 2 || ( G - E ) ) )
2	1	simpld 465	. . . . 5 |- ( ph -> E < G )
3	2	adantr 471	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> E < G )
4		signslema.4	. . . . . . . . . 10 |- ( ph -> H e. NN0 )
5	4	nn0cnd 10916	. . . . . . . . 9 |- ( ph -> H e. CC )
6		signslema.2	. . . . . . . . . 10 |- ( ph -> F e. NN0 )
7	6	nn0cnd 10916	. . . . . . . . 9 |- ( ph -> F e. CC )
8	5, 7	subcld 9972	. . . . . . . 8 |- ( ph -> ( H - F ) e. CC )
9		signslema.3	. . . . . . . . . 10 |- ( ph -> G e. NN0 )
10	9	nn0cnd 10916	. . . . . . . . 9 |- ( ph -> G e. CC )
11		signslema.1	. . . . . . . . . 10 |- ( ph -> E e. NN0 )
12	11	nn0cnd 10916	. . . . . . . . 9 |- ( ph -> E e. CC )
13	10, 12	subcld 9972	. . . . . . . 8 |- ( ph -> ( G - E ) e. CC )
14	8, 13	subeq0ad 9982	. . . . . . 7 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 0 <-> ( H - F ) = ( G - E ) ) )
15	14	biimpa 491	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( H - F ) = ( G - E ) )
16	15	breq2d 4385	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( 0 < ( H - F ) <-> 0 < ( G - E ) ) )
17	6	nn0red 10915	. . . . . . 7 |- ( ph -> F e. RR )
18	4	nn0red 10915	. . . . . . 7 |- ( ph -> H e. RR )
19	17, 18	posdifd 10188	. . . . . 6 |- ( ph -> ( F < H <-> 0 < ( H - F ) ) )
20	19	adantr 471	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( F < H <-> 0 < ( H - F ) ) )
21	11	nn0red 10915	. . . . . . 7 |- ( ph -> E e. RR )
22	9	nn0red 10915	. . . . . . 7 |- ( ph -> G e. RR )
23	21, 22	posdifd 10188	. . . . . 6 |- ( ph -> ( E < G <-> 0 < ( G - E ) ) )
24	23	adantr 471	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( E < G <-> 0 < ( G - E ) ) )
25	16, 20, 24	3bitr4rd 294	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( E < G <-> F < H ) )
26	3, 25	mpbid 215	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> F < H )
27		0red 9630	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 e. RR )
28	22, 21	resubcld 10035	. . . . . 6 |- ( ph -> ( G - E ) e. RR )
29	28	adantr 471	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( G - E ) e. RR )
30	18, 17	resubcld 10035	. . . . . 6 |- ( ph -> ( H - F ) e. RR )
31	30	adantr 471	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( H - F ) e. RR )
32	2	adantr 471	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> E < G )
33	23	adantr 471	. . . . . 6 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( E < G <-> 0 < ( G - E ) ) )
34	32, 33	mpbid 215	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 < ( G - E ) )
35		2pos 10689	. . . . . . 7 |- 0 < 2
36		breq2 4377	. . . . . . 7 |- ( ( ( H - F ) - ( G - E ) ) = 2 -> ( 0 < ( ( H - F ) - ( G - E ) ) <-> 0 < 2 ) )
37	35, 36	mpbiri 241	. . . . . 6 |- ( ( ( H - F ) - ( G - E ) ) = 2 -> 0 < ( ( H - F ) - ( G - E ) ) )
38	28, 30	posdifd 10188	. . . . . . 7 |- ( ph -> ( ( G - E ) < ( H - F ) <-> 0 < ( ( H - F ) - ( G - E ) ) ) )
39	38	biimpar 492	. . . . . 6 |- ( ( ph /\ 0 < ( ( H - F ) - ( G - E ) ) ) -> ( G - E ) < ( H - F ) )
40	37, 39	sylan2 481	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( G - E ) < ( H - F ) )
41	27, 29, 31, 34, 40	lttrd 9782	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> 0 < ( H - F ) )
42	19	adantr 471	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( F < H <-> 0 < ( H - F ) ) )
43	41, 42	mpbird 240	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> F < H )
44	5, 10, 7, 12	sub4d 10021	. . . . 5 |- ( ph -> ( ( H - G ) - ( F - E ) ) = ( ( H - F ) - ( G - E ) ) )
45		signslema.6	. . . . 5 |- ( ph -> ( ( H - G ) - ( F - E ) ) e. { 0 , 2 } )
46	44, 45	eqeltrrd 2530	. . . 4 |- ( ph -> ( ( H - F ) - ( G - E ) ) e. { 0 , 2 } )
47		ovex 6303	. . . . 5 |- ( ( H - F ) - ( G - E ) ) e. _V
48	47	elpr 3953	. . . 4 |- ( ( ( H - F ) - ( G - E ) ) e. { 0 , 2 } <-> ( ( ( H - F ) - ( G - E ) ) = 0 \/ ( ( H - F ) - ( G - E ) ) = 2 ) )
49	46, 48	sylib 201	. . 3 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 0 \/ ( ( H - F ) - ( G - E ) ) = 2 ) )
50	26, 43, 49	mpjaodan 800	. 2 |- ( ph -> F < H )
51	1	simprd 469	. . . . 5 |- ( ph -> -. 2 || ( G - E ) )
52	51	adantr 471	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> -. 2 || ( G - E ) )
53	15	breq2d 4385	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> ( 2 || ( H - F ) <-> 2 || ( G - E ) ) )
54	52, 53	mtbird 307	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 0 ) -> -. 2 || ( H - F ) )
55		2z 10958	. . . . . . 7 |- 2 e. ZZ
56	9	nn0zd 11027	. . . . . . . 8 |- ( ph -> G e. ZZ )
57	11	nn0zd 11027	. . . . . . . 8 |- ( ph -> E e. ZZ )
58	56, 57	zsubcld 11034	. . . . . . 7 |- ( ph -> ( G - E ) e. ZZ )
59		dvdsaddr 14354	. . . . . . 7 |- ( ( 2 e. ZZ /\ ( G - E ) e. ZZ ) -> ( 2 || ( G - E ) <-> 2 || ( ( G - E ) + 2 ) ) )
60	55, 58, 59	sylancr 674	. . . . . 6 |- ( ph -> ( 2 || ( G - E ) <-> 2 || ( ( G - E ) + 2 ) ) )
61	51, 60	mtbid 306	. . . . 5 |- ( ph -> -. 2 || ( ( G - E ) + 2 ) )
62	61	adantr 471	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> -. 2 || ( ( G - E ) + 2 ) )
63		2cnd 10670	. . . . . . 7 |- ( ph -> 2 e. CC )
64	8, 13, 63	subaddd 9990	. . . . . 6 |- ( ph -> ( ( ( H - F ) - ( G - E ) ) = 2 <-> ( ( G - E ) + 2 ) = ( H - F ) ) )
65	64	biimpa 491	. . . . 5 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( ( G - E ) + 2 ) = ( H - F ) )
66	65	breq2d 4385	. . . 4 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> ( 2 || ( ( G - E ) + 2 ) <-> 2 || ( H - F ) ) )
67	62, 66	mtbid 306	. . 3 |- ( ( ph /\ ( ( H - F ) - ( G - E ) ) = 2 ) -> -. 2 || ( H - F ) )
68	54, 67, 49	mpjaodan 800	. 2 |- ( ph -> -. 2 || ( H - F ) )
69	50, 68	jca 539	1 |- ( ph -> ( F < H /\ -. 2 || ( H - F ) ) )

Syntax hints:   -. wn 3   -> wi 4   <-> wb 189   \/ wo 374   /\ wa 375   = wceq 1447   e. wcel 1890   {cpr 3937   class class class wbr 4373  (class class class)co 6275   RRcr 9524   0cc0 9525   + caddc 9528   < clt 9661   - cmin 9846   2c2 10647   NN0cn0 10858   ZZcz 10926   || cdvds 14315

This theorem was proved from axioms:  ax-mp 5  ax-1 6  ax-2 7  ax-3 8  ax-gen 1672  ax-4 1685  ax-5 1761  ax-6 1808  ax-7 1854  ax-8 1892  ax-9 1899  ax-10 1918  ax-11 1923  ax-12 1936  ax-13 2091  ax-ext 2431  ax-sep 4496  ax-nul 4505  ax-pow 4553  ax-pr 4611  ax-un 6570  ax-resscn 9582  ax-1cn 9583  ax-icn 9584  ax-addcl 9585  ax-addrcl 9586  ax-mulcl 9587  ax-mulrcl 9588  ax-mulcom 9589  ax-addass 9590  ax-mulass 9591  ax-distr 9592  ax-i2m1 9593  ax-1ne0 9594  ax-1rid 9595  ax-rnegex 9596  ax-rrecex 9597  ax-cnre 9598  ax-pre-lttri 9599  ax-pre-lttrn 9600  ax-pre-ltadd 9601  ax-pre-mulgt0 9602

This theorem depends on definitions:  df-bi 190  df-or 376  df-an 377  df-3or 987  df-3an 988  df-tru 1450  df-ex 1667  df-nf 1671  df-sb 1801  df-eu 2303  df-mo 2304  df-clab 2438  df-cleq 2444  df-clel 2447  df-nfc 2581  df-ne 2623  df-nel 2624  df-ral 2741  df-rex 2742  df-reu 2743  df-rab 2745  df-v 3014  df-sbc 3235  df-csb 3331  df-dif 3374  df-un 3376  df-in 3378  df-ss 3385  df-pss 3387  df-nul 3699  df-if 3849  df-pw 3920  df-sn 3936  df-pr 3938  df-tp 3940  df-op 3942  df-uni 4168  df-iun 4249  df-br 4374  df-opab 4433  df-mpt 4434  df-tr 4469  df-eprel 4722  df-id 4726  df-po 4732  df-so 4733  df-fr 4770  df-we 4772  df-xp 4817  df-rel 4818  df-cnv 4819  df-co 4820  df-dm 4821  df-rn 4822  df-res 4823  df-ima 4824  df-pred 5358  df-ord 5404  df-on 5405  df-lim 5406  df-suc 5407  df-iota 5524  df-fun 5562  df-fn 5563  df-f 5564  df-f1 5565  df-fo 5566  df-f1o 5567  df-fv 5568  df-riota 6237  df-ov 6278  df-oprab 6279  df-mpt2 6280  df-om 6680  df-wrecs 7014  df-recs 7076  df-rdg 7114  df-er 7349  df-en 7556  df-dom 7557  df-sdom 7558  df-pnf 9663  df-mnf 9664  df-xr 9665  df-ltxr 9666  df-le 9667  df-sub 9848  df-neg 9849  df-nn 10598  df-2 10656  df-n0 10859  df-z 10927  df-dvds 14316

This theorem is referenced by: (None)